Since I don't prefer any of the other solutions since they don't have the specific features I want, [I made my own][2] (currently command-line only) program specifically for quickly capturing .gifs and .webm files for the web.

Features
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* Follows the mouse or captures fullscreen
* You can specify width and height of a recording area around your mouse
* Outputs to (lots of png files) .gif and .webm

![A gif made with screencapture][1]

[It's available for download here.][2]

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*Note that this was made in a day, if you're not a developer, you probably won't be able to install/use it yet. I will be updating this program to make it more user friendly*

  [1]: https://i.sstatic.net/qRsmy.gif
  [2]: https://github.com/Azeirah/screencap